How should I model all available actions of a chess game in deep Q-learning?
To model chess as a Markov decision problem (MDP) you can refer to the AlphaZero paper (Mastering Chess and Shogi by Self-Play with a General Reinforcement Learning Algorithm). The exact details can be found starting from the bottom of page 13.

Briefly, an action is described by picking a piece and then picking a move with it. The size of the board is 8 by 8 so there can be 8x8 possibilities for picking a piece. Then we can either pick linear movements (in 8 directions) and then pick the number of steps in that direction (maximum 7 steps) or we can make a knight movement (maximum 8 possibilities). So far that is 8x7 + 8. Furthermore, we also need to consider underpromotions (promoting a pawn into a non-queen piece). In this scenario we can have 3 types of pawn movements (forward, left diagonal or right diagonal capture) and 3 types of promotions (rook, knight, bishop) so that makes it 9. So the total dimension of the action space is 8x8x(8x7+8+9) and this will be the number of neuron outputs you will need to use.

Note that this action space representation covers every possible scenario and for example at the start of the game the action of picking the tile E4 and promoting it to a bishop doesn't make sense (there are no pieces on tile E4 at the beginning of the game). Or if we pick a tile where there is a rook we cannot make a knight movement with it. Therefore you will also need to implement a function that can return you the set of possible actions in a given state and ignore all neural network outputs that is not contained in this set.

Obviously this action representation is not set into stone so if you can come up with something better or more compact you can use that one too. You can also make restrictions to your game for example by not allowing underpromotions.